Pries: 676 Number Theory. 2010. Homework 9.
Zeta Functions
1. Use the Euler product for ζ(s) to show that:
A. There are infinitely many primes.
P
B. The sum p p−1 over all primes diverges.
C. There are no zeros ζ(s) 6= 0 if Re(s) > 1.
2. A result due to Euler:
A. Ignoring all issues of convergence, show that:
sin(x) = Cx
∞
Y
m=1
(1 −
x2
).
m2 π 2
B. Explain why C = 1.
C. Equate the coefficients of x3 to show ζ(2) = π 2 /6.
3. Prove Dirichlet’s Theorem on Arithmetic Progressions for m = 3. In other words, show
that the density of P = {p| p ≡ 1 mod 3} is 1/2 using the Dirichlet L-function for χ3 .
√
4. Let K = Q( −3). Express the Dedekind zeta-function ζK (s) in terms of ζ(s) and a
Dirichlet L-function. Write out the first few terms of ζK (s). Find the number of ideals
in the integral closure of K with norm 300.
5. Write an outline for a 2 page paper/15 minute final presentation.